Linear Equations are tools for problem solving. We represent unknown quantities in a problem by using variables such as x, y, and z depending on the number of unknown quantities. Accordingly we have linear equations in one, two or three variables.We solve the linear equations to find the unknown quantities.
Let us see some problems
1. Sheela is twice as old as her brother. If the sum of their ages is 24 years, find their ages.
Let brother’s age be x . Now Sheela’s age will be 2x.
As per the condition, sum of their ages = 24 years
Thus 2x + x = 24 is the required equation.
or, 3x=24
This is an example of a linear equation in one variable x.
In order to solve this equation, we can do one or more of the following operations.
i) We can add the same number to both sides of the equation
ii) We can subtract the same number from both sides of the equation
iii) We can multiply both sides of the equation with the same number
iv) We can divide both sides of the equation by the same number.
In this problem, we divide both sides by the same number 3 to get the value of x
3x/3 = 24/3
x = 8
Therefore, brother’s age = 8 years and Sheela’s age= 16 years.
Now let us see how to solve the same problem by taking two variables.
Let brother’s age be x and Sheela’s age be y
As per the first condition, Sheela’s age = 2 times brother’s age
i.e. y = 2x …..equation (1)
As per the second condition, sum of their ages = 24 years
i.e. x + y = 24 ….. equation (2
Substitute the value of y from equation (1) in equation (2), we get
x + 2x =24
or, 3x = 24
Now we divide both sides by 3 to get the value of x
3x/3 = 24/3
x = 8
when we replace x by its value in the equation y = 2x, we get
y = 16
Therefore, brother’s age = 8 years and Sheela’s age= 16 years.
2. Sum of the digits of a two digit number is 7. When we interchange the digits, it is found that the resulting new number is greater than the original number by 9. What is the two digit number?
Let the units digit be x and the tens digit be y.
The sum of the digits is 7
Therefore, x + y = 7 ……equation (1)
The original number is 10y + x
The number with reversed digits will be 10x + y
As per the second condition, The number with the reversed digits = The original number + 9
Therefore, 10x + y =10y + x + 9
on simplification, we get
x – y = 1 …. equation (2)
Now let us solve
x + y = 7 ……equation (1)
x – y = 1 ……. equation (2)
We can eliminate y by adding equations (1) and (2)
2x = 8
Dividing both the sides by 2, we get
2x/2 = 8/2
or, x = 4
Now we replace x by its value in equation (1), we get
4 + y = 7
Subtracting 4 from both the sides of the equation, we get
4 – 4 + y = 7 – 4
or, y = 3
Therefore, the number is
10×3 + 4 = 34.
We can check the result by adding 9 to this number to get 43, which is the number obtained by reversing the digits!
Are you ready to solve a problem?
2 pens and 3 pencils cost $12 and 3 pens and 2 pencils cost $13. Find the cost of 5 pens and 5 pencils.
Answer: $25
Hope you enjoyed the lesson on Linear Equations.
